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Difference between revisions of "Equivalence relation"

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Let <math>S</math> be a set.  A relation <math>\sim</math> on <math>S</math> is said to be an '''equivalence relation''' if <math>\sim</math> satisfies the following three properties:
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Let <math>S</math> be a [[set]].  A [[binary relation]] <math>\sim</math> on <math>S</math> is said to be an '''equivalence relation''' if <math>\sim</math> satisfies the following three properties:
  
 
1. For every element <math>x \in S</math>, <math>x \sim x</math>.  (Reflexive property)
 
1. For every element <math>x \in S</math>, <math>x \sim x</math>.  (Reflexive property)
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3. If <math>x, y, z \in S</math> such that <math>x \sim y</math> and <math>y \sim z</math>, then we also have <math>x \sim z</math>.  (Transitive property)
 
3. If <math>x, y, z \in S</math> such that <math>x \sim y</math> and <math>y \sim z</math>, then we also have <math>x \sim z</math>.  (Transitive property)
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Some common examples of equivalence relations:
 
Some common examples of equivalence relations:
  
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* The relation <math>=</math> (equality), on the set of real numbers.
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* The relation <math>=</math> (equality), on the set of [[real number]]s.
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* The relation <math>\cong</math> (congruence), on the set of geometric figures in the plane.
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* The relation <math>\cong</math> (congruence), on the set of geometric figures in the [[plane]].
 
* The relation <math>\sim</math> (similarity), on the set of geometric figures in the plane.
 
* The relation <math>\sim</math> (similarity), on the set of geometric figures in the plane.
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* For a given positive integer <math>n</math>, the relation <math>\equiv</math> (mod <math>n</math>), on the set of integers.  ([[Modular arithmetic|Congruence mod n]])
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* For a given [[positive integer]] <math>n</math>, the relation <math>\equiv \pmod n</math>, on the set of [[integer]]s.  ([[Congruence]] [[Modular arithmetic|mod ''n'']])

Revision as of 12:02, 3 August 2006

Let $S$ be a set. A binary relation $\sim$ on $S$ is said to be an equivalence relation if $\sim$ satisfies the following three properties:

1. For every element $x \in S$, $x \sim x$. (Reflexive property)

2. If $x, y \in S$ such that $x \sim y$, then we also have $y \sim x$. (Symmetric property)

3. If $x, y, z \in S$ such that $x \sim y$ and $y \sim z$, then we also have $x \sim z$. (Transitive property)


Some common examples of equivalence relations:

  • The relation $=$ (equality), on the set of real numbers.
  • The relation $\cong$ (congruence), on the set of geometric figures in the plane.
  • The relation $\sim$ (similarity), on the set of geometric figures in the plane.
  • For a given positive integer $n$, the relation $\equiv \pmod n$, on the set of integers. (Congruence mod n)
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